Recall the construction of the reduced crossed product:

Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra with an action $\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the $*$-algebra $C_c(\Gamma,A)$ of finitely supported functions $\Gamma \to A$ with the $\alpha$-twisted multiplication and involution. Then we can build a canonical faithful representation of $C_c(\Gamma,A)$ as follows: start with a faithful representation $A \subseteq B(H)$. This induces a new faithful representation $\pi: A \to B(H \otimes \ell^2(\Gamma))$ by $\pi(a)(\xi \otimes \delta_g) = \alpha_{g}^{-1}(a)\xi \otimes \delta_g$. Considering the left regular representation $\lambda: \Gamma \to U(\ell^2(H)): g \mapsto (\delta_h \mapsto \delta_{gh})$, we obtain an induced faithful representation $$C_c(\Gamma,A) \to B(H \otimes \ell^2(\Gamma)): \sum_{s \in \Gamma} a_s s \mapsto \sum_{s \in \Gamma} \pi(a_s)(1 \otimes \lambda_s)$$ which induces a $C^*$-norm on $C_c(\Gamma,A)$. The reduced crossed product $A \rtimes_r \Gamma$ is the $C^*$-completion of $C_c(\Gamma,A)$ with respect to this norm, and does not depend on the choice of faithful representation $A\subseteq B(H)$.

Let $\Gamma$ be a discrete group and let $\varphi: A \to B$ be a $\Gamma$-equivariant completely positive contraction between the $\Gamma$-$C^*$-algebras $A$ and $B$. I want to show the following:

The induced map $C_c(\Gamma,A) \to C_c(\Gamma,B): \sum_{s \in \Gamma} a_s s \mapsto \sum_{s \in \Gamma}\varphi(a_s)s$ is bounded, hence extends uniquely to a map $\varphi \rtimes_r \Gamma: A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$

**Attempt:** Let $\pi_A: A \to B(H_A \otimes \ell^2(\Gamma))$ and $\pi_B: B \to B(H_B \otimes \ell^2(\Gamma))$ be faithful representations as above. Then by the $C^*$-identity.
$$\|\sum_s \varphi(a_s)s\|^2 = \|\sum_s \pi_B(\varphi(a_s))(1 \otimes \lambda_s)\|^2$$
$$=\|\sum_{s,t} (1 \otimes \lambda_{s^{-1}}) \pi_B(\varphi(a_s^*)\varphi(a_t)) (1\otimes \lambda_t)\|.$$

This looks like something we could apply Cauchy-Schwarz for completely positive maps on, but the surrounding factors $1 \otimes \lambda_{s^{-1}}$ and $1 \otimes \lambda_t$ complicate this. Maybe I need to apply Cauchy-Schwarz on some carefully crafted matrix. Does anybody see how I can continue?

Of course, other approaches are also welcome! I am also interested in the following: if $\varphi$ is injective, then is the extension $A \rtimes_r \Gamma \to B \rtimes_r \Gamma$ also injective?